Which of the following functions have graphs that contain no asymptotes? Select all that apply. (2 points)

a. power; y=x^n
b. reciprocal; y=1/x
c. exponential; y=b^x
d. logarithmic; y=log b x
e. root; y=n(rt(x))

Please help I'm really confused. Thank you!

Ah, no need to be confused! I'm here to help and make things a little lighter. Let's go through each option and see if we can find any asymptotes.

a. Power function (y = x^n): This function can have asymptotes depending on the value of n. For example, if n is an odd whole number, there won't be any asymptotes. But if n is an even whole number, there could be vertical asymptotes.

b. Reciprocal function (y = 1/x): This one definitely has an asymptote. It's a vertical line at x = 0, since you can't divide by zero. So, there is an asymptote here.

c. Exponential function (y = b^x): No asymptotes to worry about here! The graph of an exponential function just keeps growing or decaying without any vertical or horizontal asymptotes.

d. Logarithmic function (y = log_b(x)): Similar to the reciprocal function, logarithmic functions have an asymptote. This time, it's a vertical line at x = 0. So, there is an asymptote here as well.

e. Root function (y = n√(x)): This function has no asymptotes either. The graph of a root function just keeps going without any vertical or horizontal asymptotes.

So, to summarize, the functions that have no asymptotes are a (power function) and e (root function). Keep smiling!

To determine which of the given functions have graphs that contain no asymptotes, we need to consider the behavior of these functions as x approaches infinity or negative infinity.

a. Power function: y = x^n
The graph of the power function can have asymptotes depending on the value of n. If n is an even number, the graph will have no asymptotes. However, if n is an odd number, the graph will have vertical asymptotes when x approaches negative infinity or positive infinity.

b. Reciprocal function: y = 1/x
The graph of the reciprocal function has a vertical asymptote at x = 0, where the function is undefined. Therefore, this function does have an asymptote.

c. Exponential function: y = b^x
The graph of the exponential function does not have any horizontal or vertical asymptotes. It will approach positive infinity as x approaches positive infinity and approach zero as x approaches negative infinity.

d. Logarithmic function: y = log_b x
The graph of the logarithmic function has a vertical asymptote at x = 0, where the function is undefined. Therefore, this function does have an asymptote.

e. Root function: y = n(rt(x))
The graph of the root function can have asymptotes depending on the value of n. If n is an even number, the graph will have no asymptotes. However, if n is an odd number, the graph will have a vertical asymptote at x = 0 where the function is undefined.

In summary, the functions that do not have asymptotes are:
a. Power function; y = x^n (when n is an even number)
c. Exponential function; y = b^x
e. Root function; y = n(rt(x)) (when n is an even number)

To determine which functions have graphs that contain no asymptotes, we need to understand what an asymptote is.

An asymptote is a line that a graph approaches but never touches or crosses. It represents a limit that the function approaches as the input approaches a certain value.

Let's go through each function and check for asymptotes:

a. Power function: y = x^n

Power functions can have asymptotes when either n is negative or when the degree of the numerator is less than the degree of the denominator. For example, the function y = 1/x has an asymptote at x = 0. However, the given form for the power function, y = x^n, does not have any restrictions, so it does not have an asymptote.

b. Reciprocal function: y = 1/x

The reciprocal function, y = 1/x, has an asymptote at x = 0 because division by zero is undefined. So, the reciprocal function has an asymptote.

c. Exponential function: y = b^x

Exponential functions, such as y = b^x, do not have asymptotes. The graph of an exponential function can approach x = -∞ or x = +∞, but it never touches or crosses a horizontal line.

d. Logarithmic function: y = log_b(x)

Logarithmic functions, such as y = log_b(x), do not have asymptotes. The graph of a logarithmic function can approach x = 0 or x = -∞, but it never touches or crosses a vertical line.

e. Root function: y = n√(x)

Root functions, such as y = n√(x), do not have asymptotes. The graph of a root function can approach x = 0, but it never touches or crosses a vertical line.

So, the functions that have graphs that contain no asymptotes are:

a. Power function: y = x^n
c. Exponential function: y = b^x
d. Logarithmic function: y = log_b(x)
e. Root function: y = n√(x)

Therefore, the correct answers are a, c, d, and e.

polynomials and roots have no asymptotes.

Surely your algebra book shows the graphs of all of these.